G 2 3 because of the constraints imposed by choice of

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Unformatted text preview: Statistical Tests
The Eﬀect of Duplicate policies
67/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Statistical Tests Statistical Tests (revisited) I
The following six tests can be used to test the adherence of a
graduation to data:
chi-square test
standardised deviations test
sign test
cumulative deviation test
grouping of signs test
serial correlations test. Degrees of freedom in the Chi-square test to test the
adherence to data of a graduation depends on the method of
graduation:
lose one degree of freedom for each parameter ﬁtted when
using a parametric formula
67/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Statistical Tests Statistical Tests (revisited) II when graduating using a standard table
lose one degree of freedom for each parameter ﬁtted
also lose some degrees of freedom (e.g. 2 - 3 ) because of the
constraints imposed by choice of the standard table for graphical graduation it is diﬃcult to determine the degrees
of freedom lost - a rule of thumb is 2 or 3 degrees of freedom
for every ten or so ages 68/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Statistical Tests Tests: Comparing an experience with a standard table
s
Notations: rates available from standard tables are qx or µs + 1
x
2 Null hypothesis H0 : the true mortality rates are same as those
in the standard tables
All the following six tests can be used to test goodness of ﬁt
of a set of estimated rates against a standard table (use the
standard table rates to calculate the expected number of
deaths):
chi-square test, standardised deviations test, sign test,
cumulative deviation test, grouping of signs test, serial
correlations test.
Testing goodness of ﬁt but not under or over graduation
For the χ2 -test, the statistic is a χ2 random variable with the
degree of freedom equal the no. of age groups.
69/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
The Eﬀect of Duplicate policies 1 Introduction
2 Testing smoothness
3 Statistical tests
Preliminaries
Chi-square (χ2 ) test
Standardised Deviations Test
Signs test
Cumulative Deviations Test
Grouping of Signs Test
Serial Correlations Test
4 Methods of graduation
Preliminaries
Graduation by Parametric Formula
Graduation by Reference to a Standard Table
Graphical graduation
Statistical Tests
The Eﬀect of Duplicate policies
70/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
The Eﬀect of Duplicate policies The Eﬀect of Duplicate policies
For insured lives investigations we observe policy years and
numbers of policies that are death claims and not the lives
This gives rise to duplicate policies and the assumption of
independence does not hold
Assume N lives from age x to x + 1
Assume proportion πi lives own i insurance policies (these
proportions are unknown)
Total number of policies is
i πi N
i 70/72 Assume the mortality rate for each life is qx
Let Di be the number of deaths among the πi N lives each
with i policies and Ci be the number of claims among these
lives Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
The Eﬀect of Duplicate policies Assuming the Binomial model, we have
Di ∼ Bin (πi N , qx )
We then have
E [C ] = E Ci = E iDi i = i iE [Di ] = i πi Nqx i i And
var [C ] = var Ci = var
i i 2 var [Di ] = =
i
71/72 iDi
i i 2 πi Nqx (1 − qx )
i Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
The Eﬀect of Duplicate policies If all
have i i πi N death claims were independent then we would
E [C ] = i πi N qx i var [C ] = i πi N qx (1 − qx ) i Duplicate policies increase the variance of the number of
claims by the ratio
2
i i πi
i i πi 72/72 which diﬀers for each age
We could make allowance for the increased variances in
statistical tests, if the variance ratios are known
The CMIB estimates these ratios: typically between 1.18 to
1.75, average 1.46...
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